Report¶

Group name: Gruppe B Raghid Allaham & Niels-Janos Simmet

Introduction¶

Over the last couple of years American Football gained a lot of popularity in Europe especially Germany. Different than soccer, American football is a play based game. After each play the game gets interupted for the offense and defense to set up their team once again. The goal is to achieve a touchdown on the opponents endzone. To achieve that the offense works its way across the field play by play. With each play the offense tries to gain as many yards as possible while the defense tries to prevent this. There are 2 main kind of plays to gain yards: rushing and passing. For a rush plays the offensive team tries to gain yards by caring the ball. For pass plays the quarterback of the offensive team will throw to ball to another player. The key difference is the that for rushing plays the ball will not be airborn but handed over.

We want to find out if there is a way to predict how many yards a team will gain in a play based on variables that we know before the play starts.

Other people have already researched similiar questions but i.e. focused on predicting the rushing yards. We focus on yards generally gained no matter if rushed or passed, as this can not be determined before the play starts. Interesting links to check out for similiar analysis:

http://cs229.stanford.edu/proj2019aut/data/assignment_308832_raw/26588266.pdf

https://rpubs.com/woutcault/Final606_Multiple_Regression

https://medium.com/@matthewdmeans/predicting-yards-gained-in-the-nfl-3a0eea7a54a3

Data Description¶

For NFL matches there is a comprehensive data asset available that includes play-by-play data, we will focus on the data gathered so far (until 26th December 2022) in the 2022 NFL season. It tracks a multitude of variables for each play. For our analysis we will focus on the variables that are known before the play starts as explanatory variables and the yards gained as the response variable. The data was scraped from the NFL Next Gen Stats website with the NFL verse package.

Data Source https://github.com/nflverse/nflverse-data/releases/tag/pbp

Data Description https://mrcaseb.github.io/pages_dummy/reference/fast_scraper.html

Our response variable will be "yards-gained" this variable describes how many yards a team gained (or lost) in a play excluding yards gained via fumble recoveries and laterals. Our possible explanatory variables can be found in the data dictionary below.

Data Dictionary¶

	Name	Description	Role	Type	Format
0	yards_gained	Yards gained/lost in the play excluding yards ...	Response	numeric	float64
1	quarter_seconds_remaining	Seconds on the clock until the quarter ends	Predictor	numeric	int64
2	qtr	Current quater 1-4 and 5 for overtime	Predictor	numeric	int64
3	down	Current down	Predictor	numeric	float64
4	yardline_100	Distance to the opponents endzone in yards	Predictor	numeric	float64
5	ydstogo	Yards to the next first down	Predictor	numeric	int64
6	score_differential	Score difference between offense team and the ...	Predictor	numeric	float64
7	posteam	Offense Team	Predictor	nominal	object
8	defteam	Defense Team	Predictor	nominal	object
9	home_team	Home Team	Predictor	nominal	object
10	away_team	Guest/Away Team	Predictor	nominal	object

Data Setup¶

Import data¶

We import the complete data set which also includes variables which can only be gathered after the play i.e. if a rush or a pass play was tried.

Data corrections¶

As we want to find out if there is a way to predict how many yards a team will gain based on variables that we know before the play starts, we reduce the variables to the following small list which represents the variables, that are already known before a play. We also drop all plays during which one of the variables is not available.

Data structure¶

<class 'pandas.core.frame.DataFrame'>
Int64Index: 35058 entries, 2 to 42322
Data columns (total 11 columns):
 #   Column                     Non-Null Count  Dtype  
---  ------                     --------------  -----  
 0   yards_gained               35058 non-null  float64
 1   home_team                  35058 non-null  object 
 2   away_team                  35058 non-null  object 
 3   quarter_seconds_remaining  35058 non-null  int64  
 4   qtr                        35058 non-null  int64  
 5   down                       35058 non-null  float64
 6   yardline_100               35058 non-null  float64
 7   ydstogo                    35058 non-null  int64  
 8   posteam                    35058 non-null  object 
 9   defteam                    35058 non-null  object 
 10  score_differential         35058 non-null  float64
dtypes: float64(4), int64(3), object(4)
memory usage: 3.2+ MB

Variable lists¶

We differentiate between categorial variables and numerical variables.

Data preparation¶

Next we split our data into explanatory variables and the response variables.

For the categorial variables we need to create dummy columns to fit them into our regression models.

Data splitting¶

We do a train/test split with 70/30 % of the data.

Analysis¶

Descriptive statistics¶

yards_gained                   4.668435
quarter_seconds_remaining    424.119516
qtr                            2.574391
down                           1.997005
yardline_100                  51.043585
ydstogo                        8.464231
score_differential            -1.414114
dtype: float64

	count	mean	std	min	25%	50%	75%	max
yards_gained	35058.0	4.668435	8.131966	-26.0	0.0	2.0	7.0	98.0
quarter_seconds_remaining	35058.0	424.119516	272.601532	0.0	174.0	413.0	659.0	900.0
qtr	35058.0	2.574391	1.135944	1.0	2.0	3.0	4.0	5.0
down	35058.0	1.997005	1.003640	1.0	1.0	2.0	3.0	4.0
yardline_100	35058.0	51.043585	24.274817	1.0	32.0	55.0	71.0	99.0
ydstogo	35058.0	8.464231	4.081100	1.0	6.0	10.0	10.0	38.0
score_differential	35058.0	-1.414114	9.296843	-37.0	-7.0	0.0	3.0	37.0

we have 35058 observations (individual plays)

We notice that the mean of yards gained after each play is about 4.6 with standard deviation of 8.1
The rest of the data only reflects the rules and characteristics of the game.

Exploratory data analysis¶

We can see that many plays result in zero yards gained. This is due to plays being aborted i.e. by incomplete passes or fouls.

After filtering out the plays that result zero yards gained, we get a distribution, that looks like a positive skewed normal distribution, with a mean = 4.6.

Probability that a team gains positive yards with a play: 57.94

Probability that a team gains positive yards with a play is 57.94 %

Relationships¶

Now we want to understand the relationship between our numerical variables.

Some relationships in the data can already be seen, most of them are game roles and restriction related i.e.

the maximum yards that can be gained are to the finish line, therefor there is a hard cut off and yards_gained can never be bigger than yardline_100
a high score_differntial is uncommon therefor some relationships look like a normal distribution i.e. yards_gained & score_differential this is probably due to the fact, that high yard gains are uncommon as well as high score differences, therefor this situations don't occure often
the score_differential becomes wider with every quarter, which makes sense, as the dominant team has more time to score points
the 5th qtr is overtime and therefor shorter, which also leaves less occassions for potential high yard plays. As can be seen the yards_gained and qtr relationship
it seems in the 4th down there are less yards_gained then in the others. Which can be explained by team going for a kick in the 4th down to put the ball deep into the opponents half but does not gain any yards.

Correlation Matrix¶

With a correlation matrix we want to test which of our variables correlate to each other by what amount. For that we chose the Spearman Method over the Pearson and Kendall Method. It does not really on normality of the data as the Pearson method does. And tests for correlation not dependence as the Kendall Method does

	quarter_seconds_remaining	qtr	down	yardline_100	ydstogo	score_differential	yards_gained
quarter_seconds_remaining	1.000000	-0.040000	-0.030000	0.080000	-0.000000	0.010000	0.010000
qtr	-0.040000	1.000000	0.020000	-0.040000	0.000000	-0.030000	-0.020000
down	-0.030000	0.020000	1.000000	-0.080000	-0.460000	-0.010000	0.050000
yardline_100	0.080000	-0.040000	-0.080000	1.000000	0.220000	-0.010000	0.090000
ydstogo	-0.000000	0.000000	-0.460000	0.220000	1.000000	-0.000000	0.090000
score_differential	0.010000	-0.030000	-0.010000	-0.010000	-0.000000	1.000000	-0.070000
yards_gained	0.010000	-0.020000	0.050000	0.090000	0.090000	-0.070000	1.000000

We can see a that most of the variables do not strongly correlate with each other.

ydstogo and down are quite strongly correlated as expected, because with each down the offensive team should come closer to the yardline it needs to reach.

ydstogo and yardline_100 also have some correlation which can be explained by the underlying rules of the game. The offensive team can not have more yards to go than they have until they enter the defenses endzone, the distance to which is represented by the yardline_100 variable. Therefor it's always ydstogo <= yardline_100

yards_gained and down also seem to have a correlation which can be explained by

the offensive team rather punting (kicking the ball to the direction of the defenses endzone which usually allows the defense to become offense) than trying to gain yards on 4th down
game tactics as the target distance ydstogo correlates with down. Teams might more often go for plays that result in just making it to the yardline on 3rd or 4th down rather than aiming for big plays that result in many yards gained.

Modeling¶

In this section we will look at 3 different regression models (Linear regression, K-nearest neighbour, & Lasso-regression) as well as Model 4 were we just always predict the mean value as a sanity check.

Model 1: Linear regression model¶

	quarter_seconds_remaining	qtr	down	yardline_100	ydstogo	...
2	896	1	1.0	78.0	10	...
3	869	1	1.0	59.0	10	...
4	865	1	2.0	59.0	10	...
5	841	1	3.0	54.0	5	...
6	833	1	4.0	64.0	15	...

5 rows × 134 columns

Forward selection¶

Because we have a lot a variables, we would like to choose the 10 most relevant variables for the model to make it more sensitive in regards to these important variables. Doing so, we use Forward selection:

LinearRegression()

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After the model training we can check the 10 most important variables and their coefficient.

	Name	Coefficient
0	Intercept	4.661
1	down	-0.896
2	yardline_100	0.031
3	score_differential	-0.028
4	posteam_buf	0.945
5	posteam_kc	1.244
6	posteam_mia	1.070
7	posteam_phi	0.914
8	posteam_sf	0.890
9	home_team_den	-1.026
10	home_team_det	0.889

Evaluation on Test set¶

63.43250919198005

7.9644528495044815

5.307034643163722

Save Model¶

'../models/linear_reg_model.sav_2023-01-16-21_38_42.sav'

Model 2: K-Nearest Neighbor Model¶

KNeighborsRegressor(n_neighbors=2)

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Evaluation on Test set¶

86.06705172085948

9.277232977610268

6.069832667807568

Save Model¶

'../models/KN_Model.sav_2023-01-16-21_38_43.sav'

Model 3: Lasso Regression¶

Standardization¶

Lasso performs best when all numerical features are centered around 0 and have variance in the same order.
If a feature has a variance that is orders of magnitude larger than others, it might dominate the objective function and make the estimator unable to learn from other features correctly as expected.
This means it is important to standardize our features.

$$z = \frac{(x- \bar x)}{s}$$

Lasso with optimal alpha¶

To find the optimal value of alpha, we use scikit learns lasso linear model with iterative fitting along a regularization path (LassoCV).
The best model is selected by cross-validation.

k-fold cross validation¶

LassoCV(cv=5, max_iter=10000, random_state=0)

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0.0009685368136113017

Lasso(alpha=0.0009685368136113017)

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Evaluation on Test set¶

R squared training set 2.55
R squared test set 2.28

63.43151846072908

7.964390652192362

5.307091618975074

'../models/lasso_reg_model.sav_2023-01-16-21_38_47.sav'

"Model 4": Always predict the mean value¶

array([4.65334148, 4.65334148, 4.65334148, ..., 4.65334148, 4.65334148,
       4.65334148])

64.91328211074314

8.056877938180715

5.451782561442774

Conclusions¶

Out of the 3 Models we evaluated only Linear regression and Lasso regression were able to beat our naive Model where we always predicted the mean value of the training set slightly.

Both Linear Regression and Lasso Regression performed essentially the same and only slightly better than the Dummy Model.

It seems that with the given variables we were not able to create a model which can predict yards gained in a football game reliable. This reflects the unpredictable nature of the game, which makes it enjoyable by millions of fans.

	Model	MSE	RMSE	MAE
0	Linear Regression	63.432509	7.964453	5.307035
1	K-nearest neighbour	86.067052	9.277233	6.069833
2	Lasso Regression	63.431518	7.964391	5.307092
3	Mean Value Model	64.913282	8.056878	5.451783